Homework3Solution - Homework 3 Problem 1 Evaluate...

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Homework 3. Problem 1. Evaluate multidimensional integral 1 ... ... NN Vd x d x = ∫∫ where the integration area is restricted by 1 0 N r r x L = ≤≤ and all x r are positive. Solution. The easiest way to solve the problem is to consider N=1,2,3 and then guess the answer by induction. For N=1 , this integral is just V 1 =L ( = L/1!) For N=2 , this integral is the area of the triangle restricted by 0<x+y<L , which is V 2 = L 2 /2 = (L 2 /2 !) For N=3 , this integral is the volume of tetrahedron restricted by 0<x+y+z<L , which is V 3 =L 3 /6 (=L 3 /3!) OK, now you got it! For general N this is going to be V N =L N /N! Problem 2. (Pathria, 2.7, see hints how to evaluate multidimensional integrals in 2.8 and also the result of the previous problem) Consider a system of N distinguishable one dimensional harmonic oscillators. Assume N is very large. 1. For a given macroenergy E of this system, find the number of microstates function Ω (E,N,V) using the result that the quantum energies for each oscillator (1 / 2 ) nv n ε ω =+ = and that E is much larger than the energy associated with zero- point motions (quasiclassical limit). 2. For a given macroenergy E of this system, find the available volume of the phase space using the result that the classical Hamiltonian for each oscillator 22 2 (,) v p mq Hpq m
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3. Establish the correspondence between two results showing that the conversion factor is indeed h N .
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This note was uploaded on 05/24/2010 for the course PHYSICS statistics taught by Professor Pathria during the Spring '09 term at Jackson Community College.

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Homework3Solution - Homework 3 Problem 1 Evaluate...

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